Definition
Given a set of k + 1 data points
where no two are the same, the interpolation polynomial in the Lagrange form is a linear combination
of Lagrange basis polynomials
Note how, given the initial assumption that no two are the same, so this expression is always well-defined. The reason pairs with are not allowed is that no interpolation function such that would exist; a function can only get one value for each argument . On the other hand, if also, then those two points would actually be one single point.
For all, includes the term in the numerator, so the whole product will be zero at :
On the other hand,
In other words, all basis polynomials are zero at, except, because it lacks the term.
It follows that, so at each point, showing that interpolates the function exactly.
Read more about this topic: Lagrange Polynomial
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